In the field of high precision sphere and asphere measurements, contemporary phase evaluation techniques yield surface error maps with sub-nm uncertainties. The measurand in these cases is the height deviation between the part under test and the ideal geometry measured along the surface normal.
The result of such measurements is usually delivered as a two-dimensional height map sampled on a regular grid. Initially, an interferometer gives the result in pixel coordinates of the acquisition camera. For visually assessing the quality of the surface under test, it is possible to leave the result in pixel coordinates and this is very often done. However, if the result is to be used as a feedback for the manufacturing process, it becomes necessary to present the height information in a form that allows for precisely associating each height sample with the affiliated location on the surface. Then, for instance, a polishing machine can be programmed to remove the excessive height from each point precisely.
Also for an exact numerical analysis of the figure errors of the part, it is often not admissible to leave the height results in pixel coordinates. Wavefront generators and imaging optics used for illuminating and imaging the part usually introduce a significant amount of distortion. For instance, consider a part with a large form deviation that is to be decomposed into Zernike coefficients. For this, it is necessary to know and compensate for the distortion or otherwise the Zernike decomposition will be inaccurate. A pure spherical aberration on the part could, for instance, crosstalk into other Zernike components simply because the computation grid is assumed to be equally spaced but in reality is a distorted projection of a regular grid on the part.
Therefore, together with the height values, the location of each pixel of the surface map on the surface under test is required. This information is referred to as the pixel mapping of the instrument.
This mapping is influenced by the geometry of the interferometer cavity as well as the optical imaging system that is used for relaying the part under test onto the instrument's camera.
Very often, the pixel mapping is known only implicitly, that is, by determining the location in pixel coordinates of a certain feature on the part whose physical dimension is known. This can, for instance, be the known diameter of an aperture that is mounted on top of the surface under test which can be found in image coordinates by image processing. Then a linear relationship of the pixel mapping is assumed to obtain the coordinate mapping over the full surface. This method can not account for a possible distortion of the imaging system.
Another possibility is to obtain the pixel mapping from a computer-modeled version of the imaging system. This approach has the advantage that effects like distortion or other aberrations of the imaging system can be taken into account. For this to work, highly accurate information about the various optical components in the system is necessary. The model approach can be combined with the image processing method of finding a metrically known feature on the part as described above.
In high precision sphere and asphere metrology, use is often made of additional calibration methods to measure the pixel mapping, including any possible distortion. This can be done, for example, by using a special test artifact as an object with clearly detectable features to characterize the coordinate mapping of the imaging optics. In certain cases, it is necessary to repeat such a coordinate calibration for various focus settings of the interferometer.
In general, such a coordinate calibration is a costly and tedious additional effort in high precision optical metrology.
Knowing the exact coordinate mapping between a camera and the part under test in ultra-precise asphere metrology, which uses a very high accuracy manipulator stage, requires knowledge of the coordinate mapping on two occasions: first; in the calibration of the reference surface, and second, during the alignment of the surface under test.
In ultra high precision metrology, it is known that the manipulator stage allows moving the part in 6 degrees of freedom with nm control over the rigid body motion of the part under test in three-dimensional space. As will be described, the coordinate mapping between the camera pixels and the part under test can be determined directly from a set of phase measurements in conjunction with use of the manipulator stage.
Accordingly, a primary object of the present invention is to provide methods and apparatus for the in-situ determination of mapping from pixel coordinates onto object coordinates and vice versa by using a manipulator stage for controlling the part position in the interferometer cavity and making a few additional phase measurements at various part positions.
Other objects of the invention will, in part, be obvious and will, in part, appear hereinafter when the detailed description is read with reference to the drawings.